Paul Yiu

Department of Mathematics

Florida Atlantic University

37th Annual Meeting of the Florida Section of MAA

University of Central Florida, Orlando,

February 25--26, 2004

- 1. Reflections and isogonal lines
- isogonal conjugates
- reflections of P and P* have congruent circumcircles
- O and H are isogonal conjugates
- reflections of H lie on the circumcircle
- common pedal circle of O and H
- nine-point circle

- 2. Examples of isogonal conjugates
- Centroid and symmedian point (perspector
of tangential triangle)

- Gergonne point and internal center of similitude of circumcircle and incircle
- Nagel point and external center of similtude of circumcircle and incircle
- isogonal conjugates of points on circumcircle

- 3. The symmedian point K: isogonal conjugate of G
- perspector of tangential triangle
- perspector of triangle bounded by outer sides of similar rectangles erected on the sides
- two Lemoine circles
- center of conic tangent to sidelines
at pedals of H

- Brocard circle (seven-point
circle)

- 4. The Euler line:
OG : GN : NH = 2:1:3

- de Longchamps point L: reflection of H in O as radical center of triad of circles A(a), B(b), C(c)
- Schiffer point Sc: common point of the Euler lines of IBC, ICA, IAB, and ABC
- circumcenter of tangential triangle
- Euler infinity point and
centroid of its cevian triangle

- 5.
Line of reflections

- Simson line and line of
reflections

- Simson lines of antipodal
points intersect orthogonally on nine-point circle

- point with given line of reflections: reflections
of line through H concur at a point on circumcircle

- reflections of Euler line concur at E on circumcircle
- circles APX, BPY, CPZ (X,Y,Z reflections
of H in AP, BP, CP) intersect on circumcircle

- 6. Some conic constructions
- tangent at a point of conic
- second intersection of line with conic
- center of conic
- de Longchamps point as the ``radical center'' of three ellipses
- Soddy circles

- 7. Rectangular hyperbolas

- Poncelet-Brianchon theorem (1822): A rectangular circum-hyperbola passes through the orthocenter and has center on the nine-point circle
- H(P) : rectangular hyperbola through P, center W(P), tangent at H
- reflections of tangent at H of rectangular circum-hyperbola through P
- asymptotes of rectangular hyperbola: regarded as infinite points, isogonal conjugates on circumcircle, antipodal. The hyperbola is the locus of isogonal conjugates of points on the circum-diameter. H(P) as isogonal conjugate of OP*, fourth common point of H(P) and circumcircle: isogonal conjugate of infinite point of OP*
- isogonal conjugate of a line: circumconic

- 8. Three examples:
- Jerabek hyperbola: isogonal conjugate of the Euler line
- Kiepert hyperbola: isogonal conjugate of OK
- Feuerbach hyperbola: isogonal conjugate of OI
- 9. Reflection conjugates

- reflection conjugate r(P) (except for H and points on the circumcircle)
- r(P)
= antipode of P in H(P), P* and (rP)* inverse in circumcircle

- 10. Inscribed conics
- inscribed conic with prescribed foci P and P*
- inscribed ellipse with foci O and H, center N
- inscribed conic
tangent to pedal circle

- construction
of inscribed conic

- 11. Inscribed parabolas
- inscribed parabola: focus F on circumcircle, directrix = line of reflections of F
- inscribed parabola tangent to a given line
- inscribed parabola tangent to Euler line
- a cubic curve: locus of P for which the inscribed parabola tangent to OP touches it at P
- 12. Inverses in circumcircle
- inversive images of traces of a point P in circumcircle
- the case of G
- P
on circumcircle: locus of perspector - isogonal conjugate of nine-point
circle

- P
on Euler line: locus of perspector - conic through the traces of the isogonal
conjugates of Kiepert and Jerabek centers

- Appendix A: reflections in altitudes
- Appendix B: barycentric
coordinates